where \(\textbf {u}\) , \(t\) , \(p\) , \(\sigma \) , \(\Gamma \) , \(C\) , and \(C_{s}\)  represent velocity, time, pressure, surface tension, interfacial surfactant concentration, bulk surfactant concentration, and bulk surfactant concentration in the vicinity of the interface, respectively. The physical parameters included in the scaling correspond to the width of the channel, \(D\) , the average inlet velocity of the liquid, \(U_{a}\) , the density of the liquid phase, \(\rho _{l}\) , the surface tension in a surfactant-free interface, \(\sigma _{s}\) , the saturation interfacial concentration, \(\Gamma _{\infty }\) , and the initial bulk surfactant concentration, \(C_{\infty }\) , in line with the scaling proposed by [1]} for a similar system. In what follows, we refer to each variable by its name to refer to its dimensionless version, unless stated otherwise and add the subscript \(b\)  to signify that the variable is reported at the bubble nose. The governing mass and momentum equations are written in dimensionless form and according to a `one-fluid' formulation, as shown in Eq. (REF )-(REF ),
\(\begin{split}\nabla \cdot \tilde{\textbf {u}}=0, ~~~~\tilde{\rho }\left(\frac{\partial \tilde{\textbf {u}}}{\partial \tilde{t}}+\tilde{\textbf {u}} \cdot \nabla \tilde{\textbf {u}}\right) &= - \nabla \tilde{p} + \frac{1}{Re}\nabla \cdot \left[ \tilde{\mu } (\nabla \tilde{\textbf {u}} +\nabla \tilde{\textbf {u}}^T) \right] \\&+ \frac{1}{ReCa} \int _{\tilde{A}(\tilde{t})} \left(\tilde{\sigma } \tilde{\kappa } \textbf {n}+\nabla _s \tilde{\sigma } \right)\delta \left(\tilde{\textbf {x}} - \tilde{\textbf {x}}_{_f} \right) d\tilde{A} ,\end{split}\) 
\( \left.\begin{array}{c}\tilde{\rho }\left( \textbf {x},t\right)=\rho _g + (\rho _{l} - \rho _{g}) \mathcal {I}\left( \textbf {x},t\right),\\\tilde{\mu }\left( \textbf {x},t\right)=\mu _{g} + (\mu _{l} - \mu _{g}) \mathcal {I}\left( \textbf {x},t\right),\\\end{array}\right.\) 
